Solving Quadratic Equation (x+3)^2=49 A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a significant position, serving as fundamental tools for modeling and solving a myriad of real-world problems. Among the diverse forms of quadratic equations, the equation (x+3)^2 = 49 presents an intriguing challenge, inviting us to embark on a journey of algebraic manipulation and problem-solving strategies. This comprehensive guide aims to dissect the intricacies of this equation, revealing its underlying structure and elucidating the step-by-step process of obtaining its solutions. By delving into the equation's core components and employing a combination of algebraic techniques, we shall unravel the values of x that satisfy the given condition, thereby gaining a deeper understanding of quadratic equations and their profound implications.
Decoding the Quadratic Equation (x+3)^2 = 49
To begin our exploration, let's first dissect the given quadratic equation, (x+3)^2 = 49, and identify its key components. This equation is expressed in the form of a squared binomial, where (x+3) represents a binomial expression that is raised to the power of 2. The equation asserts that the square of this binomial is equal to 49, a constant value. Our primary objective is to determine the values of x that, when substituted into the equation, will make the equation true. In essence, we seek to find the roots or solutions of the equation, which represent the points where the quadratic function intersects the x-axis when graphically represented. These solutions hold paramount importance as they provide insights into the behavior and properties of the quadratic function, enabling us to make informed predictions and solve related problems. The equation's structure suggests that we can employ various algebraic techniques to isolate x and determine its possible values. One approach involves expanding the squared binomial and rearranging the equation into standard quadratic form, while another method leverages the concept of square roots to directly solve for x. By carefully applying these techniques, we can systematically unravel the solutions of the equation and gain a comprehensive understanding of its mathematical properties.
Method 1: Expanding the Squared Binomial and Rearranging
The first approach to solving the quadratic equation (x+3)^2 = 49 involves expanding the squared binomial, (x+3)^2, and rearranging the equation into the standard quadratic form, ax^2 + bx + c = 0. This method provides a systematic way to tackle quadratic equations, allowing us to apply established techniques for finding solutions. Let's delve into the step-by-step process:
- Expanding the Squared Binomial: The squared binomial, (x+3)^2, can be expanded using the distributive property or the FOIL method (First, Outer, Inner, Last). Applying the distributive property, we have:
(x+3)^2 = (x+3)(x+3) = x(x+3) + 3(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9
Alternatively, using the FOIL method, we multiply the first terms (x * x = x^2), the outer terms (x * 3 = 3x), the inner terms (3 * x = 3x), and the last terms (3 * 3 = 9), and then combine the like terms to obtain the same result: x^2 + 6x + 9.
- Substituting and Rearranging: Now that we have expanded the squared binomial, we can substitute it back into the original equation:
(x+3)^2 = 49
x^2 + 6x + 9 = 49
To rearrange the equation into the standard quadratic form, ax^2 + bx + c = 0, we subtract 49 from both sides:
x^2 + 6x + 9 - 49 = 0
x^2 + 6x - 40 = 0
- Factoring the Quadratic Expression: The next step involves factoring the quadratic expression, x^2 + 6x - 40. Factoring involves finding two binomials that, when multiplied together, yield the original quadratic expression. To factor this expression, we need to find two numbers that multiply to -40 and add up to 6. These numbers are 10 and -4. Therefore, we can factor the quadratic expression as follows:
x^2 + 6x - 40 = (x + 10)(x - 4)
- Setting Factors to Zero: Now that we have factored the quadratic expression, we can set each factor to zero and solve for x:
(x + 10)(x - 4) = 0
x + 10 = 0 or x - 4 = 0
Solving for x in each equation, we get:
x = -10 or x = 4
Therefore, the solutions to the quadratic equation (x+3)^2 = 49 are x = -10 and x = 4.
Method 2: Utilizing Square Roots
An alternative method for solving the quadratic equation (x+3)^2 = 49 involves utilizing the concept of square roots. This approach offers a more direct path to the solutions, particularly when the equation is expressed in the form of a squared expression equal to a constant. Let's explore the step-by-step process:
- Taking the Square Root of Both Sides: The first step is to take the square root of both sides of the equation:
(x+3)^2 = 49
√(x+3)^2 = ±√49
Note that we include both the positive and negative square roots of 49, as both 7 and -7, when squared, yield 49. This is a crucial step in capturing all possible solutions.
- Simplifying the Equation: Simplifying the equation, we get:
x + 3 = ±7
- Isolating x: To isolate x, we subtract 3 from both sides of the equation:
x = -3 ± 7
- Determining the Solutions: Now we have two possible solutions, one with the plus sign and one with the minus sign:
x = -3 + 7 = 4
x = -3 - 7 = -10
Therefore, the solutions to the quadratic equation (x+3)^2 = 49 are x = 4 and x = -10, which are consistent with the solutions obtained using the first method.
Verifying the Solutions
To ensure the accuracy of our solutions, it is essential to verify them by substituting them back into the original equation. This process helps to confirm that the values we obtained indeed satisfy the equation and are not extraneous solutions arising from algebraic manipulations. Let's verify the solutions x = 4 and x = -10:
- Verifying x = 4: Substituting x = 4 into the original equation, we get:
(x+3)^2 = 49
(4+3)^2 = 49
7^2 = 49
49 = 49
The equation holds true, indicating that x = 4 is indeed a valid solution.
- Verifying x = -10: Substituting x = -10 into the original equation, we get:
(x+3)^2 = 49
(-10+3)^2 = 49
(-7)^2 = 49
49 = 49
The equation also holds true for x = -10, confirming that it is a valid solution as well.
Since both solutions satisfy the original equation, we can confidently conclude that x = 4 and x = -10 are the correct solutions to the quadratic equation (x+3)^2 = 49.
Conclusion
In this comprehensive guide, we have successfully unraveled the solutions of the quadratic equation (x+3)^2 = 49. By employing two distinct methods – expanding the squared binomial and utilizing square roots – we systematically derived the solutions x = 4 and x = -10. Furthermore, we verified these solutions by substituting them back into the original equation, confirming their validity. This exploration has not only provided us with the specific solutions to this particular equation but has also enhanced our understanding of quadratic equations in general. The techniques and concepts discussed here can be readily applied to solve a wide range of quadratic equations, empowering us to tackle more complex mathematical challenges. The ability to manipulate and solve quadratic equations is a fundamental skill in mathematics, with applications extending to various fields such as physics, engineering, and economics. By mastering these techniques, we equip ourselves with valuable tools for problem-solving and critical thinking, enabling us to approach real-world scenarios with greater confidence and proficiency. The journey of solving quadratic equations is not merely about finding numerical answers; it is about developing a deeper appreciation for the elegance and power of mathematical principles. As we continue our exploration of mathematics, let us embrace the challenges and strive for a comprehensive understanding of the subject, unlocking its potential to illuminate the world around us.
